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Schwarz method justification of a coupling between finite elements and integral representation for Maxwell
exterior problem
Eric Darrigrand, Nabil Gmati, Rania Rais
To cite this version:
Eric Darrigrand, Nabil Gmati, Rania Rais. Schwarz method justification of a coupling between finite
elements and integral representation for Maxwell exterior problem. Comptes Rendus. Mathématique,
Académie des sciences (Paris), 2014, 352 (4), pp.311-315. �10.1016/j.crma.2014.02.006�. �hal-00951206�
Schwarz method justification of a coupling between finite elements and integral representation for Maxwell exterior problem
Eric Darrigrand
1, Nabil Gmati
2, Rania Rais
1,21Universit´e de Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes, France
2Universit´e de Tunis El Manar, Ecole Nationale d’Ing´enieurs de Tunis, LAMSIN, B.P. 37, 1002 Le Belv´ed`ere, Tunisia Notice: this is the author’s version of a work that was accepted for publication in Comptes rendus - Math´ematique.
Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was
submitted for publication. A definitive version was subsequently published in C. R. Acad. Sci. Paris, Ser. I (2014), http://dx.doi.org/10.1016/j.crma.2014.02.006
Abstract
We are interested in the resolution of an exterior Maxwell problem in 3D using a coupling between finite elements and integral representation. This strategy is interpreted as a Schwarz method which suggests a preconditioner for Krylov solvers. Numerical results confirm the relevance of the resolution scheme.
R´esum´e
Justification par la m´ethode de Schwarz du couplage entre ´el´ements finis et repr´esentation int´egrale pour le probl`eme de Maxwell en domaine ext´erieur.Nous nous int´eressons `a la r´esolution num´erique du probl`eme de Maxwell ext´erieur en 3Dpar la m´ethode de couplage entre ´el´ements finis et repr´esentations int´egrales.
Nous limitons notre ´etude au cas d’un obstacle de type conducteur parfait. Cette strat´egie est interpr´et´ee comme une m´ethode de Schwarz qui sugg`ere un pr´econditionneur pour les solveurs de Krylov. Par suite, nous menons une ´etude num´erique pour valider la pertinence d’un tel choix.
Version fran¸caise abr´eg´ee :
Nous nous int´eressons au probl`eme de diffraction d’ondes ´electromagn´etiques par un conducteur parfait en r´egime harmonique. Pour r´esoudre ce probl`eme ext´erieur, nous consid´erons la m´ethode de couplage entre ´el´ements finis et repr´esentations int´egrales (probl`eme (2), connue sous la d´enomination CEFRI, cf. [3]). Dans ce papier, nous analysons plus sp´ecifiquement une strat´egie de r´esolution it´erative du cou- plage CEFRI inspir´ee de [4]. Nous montrons dans un premier temps l’analogie entre la m´ethode CEFRI et la m´ethode de Schwarz avec recouvrement total pour la r´esolution du probl`eme de Maxwell en domaine
Email addresses: eric.darrigrand-lacarrieu@univ-rennes1.fr(Eric Darrigrand1),nabil.gmati@enit.rnu.tn (Nabil Gmati2),rania.rais@lamsin.rnu.tn(Rania Rais1,2).
non born´e (probl`emes (5) et (6)). Cette r´einterpr´etation de la m´ethode CEFRI a ´et´e initialement pr´esent´ee dans [1]. Dans le cas d’une g´eom´etrie sph´erique, elle nous permet d’´etablir le taux de convergence de la m´ethode it´erative inh´erente au probl`eme (2) : la m´ethode de Schwarz avec recouvrement total converge si le bord Γ de l’obstacle est assez ´eloign´e de la fronti`ere artificielle Σ. Ce r´esultat est confirm´e par des tests num´eriques r´ealis´es avec des ´epaisseurs diff´erentes entre Γ et Σ (voir Fig. 2). Cette relecture de la m´ethode CEFRI offre ´egalement une technique de pr´econditionnement.
Pour des raisons pratiques de mise en œuvre num´erique, la prise en compte de la condition essentielle impos´ee sur le bord Γ, se fait par une strat´egie de p´enalisation qui consiste `a remplacer la condition au bord E×nγ = 0 du probl`eme (2) par εp (nγ ×curlE) +E×nγ = 0 o`u εp >0 est choisi petit. Apr`es discr´etisation, cette m´ethode nous am`ene `a r´esoudre un syst`eme lin´eaire impliquant une matrice ´el´ements finis classiqueAainsi qu’un bloc pleinC dˆu `a la repr´esentation int´egrale. La matrice du syst`eme lin´eaire
`
a r´esoudre A+C est mal conditionn´ee. En appliquant le pr´econditionneur sugg´er´e par la m´ethode de Schwarz sur l’´equation induite de la discr´etisation du probl`eme de diffraction (3), le syst`eme devient alors
(I+A−1C)E=A−1F, (1)
F ´etant le second membre du syst`eme initial.
Un travail num´erique de d´eveloppement de nouveaux outils int´egraux dans la librairie ´el´ements finis M´elina++([5]) a permis la validation de la m´ethode CEFRI et d’une strat´egie it´erative de type Kry- lov. Plusieurs cas tests montrent la convergence ´el´ements finis ainsi que la convergence de la r´esolution it´erative : La figure 3 (gauche) montre l’erreur relative pond´er´ee par la matrice du syst`eme lin´eaire en fonction de la finesse du maillage pour diff´erentes valeurs du param`etre de p´enalisation `a un nombre d’onde fix´e ; La figure 3 (droite) montre la convergence du solveur it´eratif pour diff´erents nombres d’onde.
Introduction
We are interested in the resolution of the 3D exterior time-harmonic Maxwell equations. To this aim, we focus on a combination of finite elements and integral representation ([3]). This strategy leads to an equivalent problem on a reduced bounded domain delimited by the surface of the scatterer and an artificial boundary with exact artificial boundary condition. No a priori condition is required on the distance between the scatterer and the artificial boundary but a difficult issue consists in the elaboration of a resolution strategy. A relevant idea was suggested in [4]. We propose the interpretation of this idea as an application of the Schwarz method, following the work done in [1] for Helmholtz equation. Hence, the theory on the Schwarz method justifies the use of Krylov solvers and the choice of a preconditioner.
In the next section, we derive the formulation of the system to be solved. Section 2 is devoted to the Schwarz interpretation of the resolution strategy suggested in [4]. In Section 3, the speed of convergence is estimated in the case of a spherical scatterer. Finally, some numerical results illustrate the theoretical properties.
1. Scattering by a perfect conductor
Let us consider Ωia bounded scatterer inR3with a regular boundary Γ and Ωeits unbounded comple- mentary. We are concerned with the scattering of a time-harmonic electromagnetic wave by the perfect conductor Ωi. Our purpose is to determine the total field E = Es+Einc where Einc is the incident wave andEsis the scattered field, solution to the regularized Maxwell equations with essential boundary
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condition on Γ and radiation condition at infinity. Considering an integral representation on an artificial
Ωi Γ Ωe
Γ Σ
Ω
Figure 1. Left: exterior unbounded domain Ωe, right: bounded domain Ω.
boundary Σ (see Fig. 1), the exterior problem reduces to a problem on a bounded domain Ω delimited by Γ and Σ (see [3]): FindE such that
curl curlE−t−1∇(divE)−ks2E= 0 in Ω, E×nγ = 0,divE= 0 on Γ,
Tν1(E) =Tν1(Einc− IΓ(E)) andNν2(E) =Nν2(Einc− IΓ(E)) on Σ,
(2)
wherenγ is the exterior unit normal of the domain Ωi on Γ; the regularization term t−1∇(divE) allows the use of a Galerkin finite-elements method (see [3]) and the regularization parameter t−1 depends on the permittivity and the permeability of the air; ks is the wave number; ν1 and ν2 are complex numbers which have a negative imaginary part. The two operatorsTν1 and Nν2 are defined by Tν1E = curlE×nσ+ν1nσ×(E×nσ) and Nν2E = divE+ν2E·nσ with nσ the exterior unit normal of the domain Ω on Σ. The boundary conditions on Σ are derived from the integral representations satisfied by the scattered field and identified by the following expression ([3]): forx∈Ωe,
IΓ(E)(x) =−ks2
Z
ΩRGt(x, .)E+ Z
Ω
curlRGt(x, .)curlE+t−1 Z
Ω
divRGt(x, .)TdivE−t−1 Z
Γ
divGt(x, .)T(E·nγ)dγ,
where Gt = GksI+ 1
k2sHess(Gks −Gkp) is the outgoing Green tensor associated with the differential operator curl curl−t−1∇(div)−ks2I of the regularized Maxwell equation;I is the identity matrix inR3; Hess stands for Hessian operator;kp=√
tksandGk is the fundamental solution of Helmholtz equation;
Ris a linear operator that maps every regular functionϕdefined on Γ into a regular functionRϕdefined on Ω which satisfiesRϕ=ϕon Γ and Rϕ= 0 on Σ. The consideration of the Hilbert space
Ht=
E∈H(curl,Ω)/divE∈L2(Ω), E×nγ = 0 on Γ, E×nσ∈L2(Σ)3, E·nσ ∈L2(Σ) , enables one to write a variational formulation of the problem (2): FindE∈ Htsuch that
(At+Ct)E=Ft, (3)
where the operatorsAtandCt:Ht→ Htare defined as follows (AtE, E′)t=
Z
Ω
(curlE·curlE′+t−1divEdivE′−ks2E·E′)+
Z
Σ
(ν1(nσ∧E)·(nσ∧E′)+t−1ν2(nσ·E)(nσ·E′))dσ,
(CtE, E′)t= Z
Σ
Tν1(IΓ(E))·E′dσ+t−1 Z
Σ
Nν2(IΓ(E))(nσ·E′)dσ, and Ft is given by (Ft, E′)t =
Z
Σ
(Tν1(Einc)·E′+t−1Nν2(Einc)(nσ·E′))dσ, where (·,·)t is the scalar product onHt.
The problem (3) is well posed as explained in [3] and the operatorAtis invertible.
2. Schwarz method interpretation
In order to solve the system (3), Jin and Liu [4] suggested to consider Ct in the right hand side. An application of the fixed point algorithm leads to findingEn+1 such that
curl curlEn+1−t−1∇(divEn+1)−k2sEn+1= 0 in Ω, En+1×nγ = 0,divEn+1= 0 on Γ,
Tν1(En+1) =Tν1(Einc− IΓ(En)) andNν2(En+1) =Nν2(Einc− IΓ(En)) on Σ.
(4)
In this paper, we focus on an original mathematical justification of convergence of the algorithm expressed by Jin and Liu. We interprete the algorithm defined by (4) as a Schwarz method. This interpretation has been initially proposed for the case of Helmholtz equation in [1]. The strategy is designed by the Total Overlapping Schwarz Method. Indeed the overlapping area is the total domain Ω. We hereby extend their work to the case of Maxwell equations: it consists in replacing equivalently the problem (4) by the two following subproblems. The first one is a transmission problem:
curl curlE2n+1−t−1∇(divE2n+1)−k2sE2n+1= 0 in Ωi∪Ωe, nγ×[E2n+1] = 0, nγ×[curlE2n+1] =−nγ×curlE2n on Γ, [divE2n+1] = 0, nγ·[E2n+1] =−nγ·E2n on Γ,
ρ→∞lim Z
||x||=ρ||curlE2n+1s ×nγ−iksnγ×(E2n+1s ×nγ)||2dγ= 0,
ρ→∞lim Z
||x||=ρ|√
t−1divE2n+1s −iksE2n+1s ·nγ|2dγ= 0.
(5)
The second one consists in findingE2n+2 such that
curl curlE2n+2−t−1∇(divE2n+2)−ks2E2n+2= 0 in Ω, E2n+2×nγ = 0, divE2n+2 = 0 on Γ,
Tν1(E2n+2) =Tν1(E2n+1) andNν2(E2n+2) =Nν2(E2n+1) on Σ.
(6)
The solutionE2n+1 of (5) has an explicit solution given by an integral representation. By inserting this representation in the second condition of (6) we effectively obtain the solution of (4). At the iterationn, the Schwarz algorithm is defined by AtEn+1 =−CtEn+Ft. Numerically, we use the scheme suggested by Jin and Liu and do not use the subproblems (5) and (6). The intermediate problems (5) and (6) are used for theoretical justifications. This enables one to derive convergence estimations that cannot be obtained directly from the system (4). In Section 3, we investigate an analytical calculation of the rate of convergence of the Schwarz method in a spherical configuration.
3. Analytical estimation of the convergence for a spherical scatterer
In the case where Ωiis a perfectly conducting ball, we investigate the rate of convergence of the Total Overlapping Schwarz method. Let us consider the scatterer to be a ball of radiusR∗. We suppose that the artificial boundary Σ is a sphere concentric to Γ with radiusR > R∗. We first introduce some notations:
We denote byjl the spherical Bessel function of degreel, byhl the spherical Hankel function of the first kind of degree l and Hl(r) =hl(r) +rh′l(r), Jl(r) = jl(r) +rj′l(r). We introduce the tangential vector
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spherical harmonics on the unit sphere S1,Ulm = (1/p
l(l+ 1))∇Ylm and Vlm =nγ ×Ulm, whereYlm, l >0,m=−l, ..., lare the orthonormal scalar spherical harmonics (complete basis ofL2(S1)). The sets of Ulm andVlm form a complete orthonormal basis forT2(S1) :=
a:S1→C3/ a∈(L2(S1))3, a·nγ= 0 . We define the error (wn)n on the fieldE at each iteration by:
w2n+1=E2n+1−Ein Ωe, w2n+1=E2n+1in Ωi, w2n+2=E2n+2−Ein Ω.
For alln, we define Λn=Tν1(w2n) andδn=Nν2(w2n). In order to prove that the error (wn)n converges to zero, we first show that Λn+1 =KΛn and δn+1=Lδn withK :T2(S1)→T2(S1) andL:L2(S1)→ L2(S1) two linear maps.K (resp.L) has a diagonal representation in the basis (Ulm, Vlm)lm ofT2(S1) (resp. Ylm of L2(S1)). Let us denote by τ1,lm, τ2,lm (resp. τ3,lm) the eigenvalues of K (resp. L). These eigenvalues define the rate of convergence of the Total Overlapping Schwarz method. Taking into account the boundary and transmission conditions, we obtain:
τ1,lm=
1−Hl(ksR∗) Jl(ksR∗)
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1
,
τ2,lm=
1−hl(ksR∗) jl(ksR∗)
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1
, τ3,lm=
1−Hl(ksR∗) Jl(ksR∗)
jl(ksR) hl(ksR)
−1
.
The convergence of the Total Overlapping Schwarz method is ensured if|τi,lm|<1,∀i= 1, ...,3,∀l. The reader can see that these eigenvalues are independent of the parameter m. ForR∗ = 1, the asymptotic behavior of the spherical Bessel functions for largelleads to the asymptotic estimationτi,lm∼(1−R2l)−1, i= 1, ...,3. As a consequence, for small values ofR, there exists a finite number of coefficientsτi,lmoutside of the unit disk, and for sufficiently large values ofR, all the coefficientsτi,lmare in the interior to the unit disk. We conclude that the Schwarz method has a linear convergence forRlarge enough. The numerical tests illustrate this theoretical result. In Fig. 2, we considerR∗= 1 andR=R∗+ewith different values ofe:λ/100,λ/10 orλ/5. The casese=λ/100 ande=λ/10 exhibit some coefficients larger than 1 while the maximum value of |τ2,lm| is strictly lower than 1 at the considered wavenumbers for the thickness e = λ/5 but the results are strongly dependent on the wavenumber. Similar asymptotic observations can be done on |τ1,lm| and |τ3,lm|. As a consequence, a Krylov method is a relevant alternative to the algorithm defined by (4): due to the properties of Krylov solvers demonstrated in [2], the convergence of a Krylov method is ensured for the resolution of the problem (3) usingAtas a preconditioner.
0 50 100 150 200 250
0 0.5 1 1.5
l (mode)
|τ2,lm|
ks=1 ks=10 ks=30
0 50 100 150 200 250
0 0.5 1 1.5
l (mode)
|τ2,lm|
ks=1 ks=10 ks=30
0 50 100 150 200 250
0 0.2 0.4 0.6 0.8
l (mode)
|τ2,lm|
ks=1 ks=10 ks=30
Figure 2. Modulus ofτ2,lmfor thicknesse=λ/100 (left),λ/10 (center) andλ/5 (right). Casesks= 1, 10 or 30.
4. Preconditioner for a Krylov solver
The previous work suggests the use ofAt as a preconditioner to solve the problem (3) using a Krylov solver. We hereby consider the resolution of problem (3) using the biconjugate gradient stabilized method.
After a finite-elements discretization, the linear system is written under the form (A+C)E=F and the
preconditioned system becomes (I+A−1C)E=A−1F where the matrixCinvolves the integral operators and the matrixArelated to the differential operators involves a term resulting from the essential condition considered by a penalization strategy:εp(nγ×curlE) +E×nγ = 0 on Γ, with εp>0.
The numerical implementation were done using and developing new integrands in the libraryM´elina++
[5]. In this section, we validate the efficiency of the preconditioning strategy by the consideration of an intermediate problem the solution of which is known:
curl curlE− ∇(divE)−k2sE= 0 in Ωe, E×nγ =G11×nγ,divE= divG11on Γ,
ρ→∞lim Z
||x||=ρ||curlEs×nγ−iksnγ×(Es×nγ)||2dγ= 0,
ρ→∞lim Z
||x||=ρ|divEs−iksEs·nγ|2dγ= 0,
(7)
whereG11is the first vector component ofG1. The scatterer is the unit sphere and the artificial boundary Σ is the sphere concentric to Γ with radiusR= 1.5.
0.1 0.15 0.2 0.25 0.3 0.35
0 0.01 0.02 0.03 0.04 0.05
h
relative l2−error
FE relative error
!p = 10−2
!p = 10−3
!p = 10−4
0 20 40 60
−15
−10
−5 0 5
iterations
log(residuals)
Bi−CGstab residuals, !
p = 10−4 ks=2.1 ks=3.1 ks=4.2
Figure 3. Relativel2-error with respect to the discretization (left), behavior of the residuals, caseksh= 2π/10 (right)
Fig. 3-left shows the convergence of the relative error with respect to the finite-elements discretization:
the relative error is plotted with respect to the average size of the mesh elements, for different values of the penalization parameter εp, with ks = 3. Fig. 3-right illustrates a superlinear convergence of the biconjugate gradient stabilized solver applied to the preconditioned system for different values of the wavenumber withεp= 10−4.
References
[1] F. Ben Belgacem, M. Fourni´e, N. Gmati, F. Jelassi, “On the Schwarz algorithms for the elliptic exterior boundary value problems”, ESAIM: M2AN, vol.39, n. 4, pp. 693-714, 2005.
[2] N. Gmati, B. Philippe, “Comments on the GMRES Convergence for Preconditioned Systems”, Large-Scale Scientific Computing, Lecture Notes in Computer Science, vol 4818, pp. 40-51, 2008.
[3] C. Hazard, M. Lenoir, “On the solution of time-harmonic scattering problems for Maxwell’s equations”, SIAM J. MATH.
ANAL., vol. 27, n. 6, pp. 1597-1630, nov. 1996.
[4] J-M. Jin, J. Liu, “A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation Problems”, IEEE Transactions on antennas and propagation, vol. 49, n. 12, pp. 1794-1806, dec. 2001.
[5] D. Martin, “http://anum-maths.univ-rennes1.fr/melina/melina++ distrib/ ”, Universit´e Rennes 1, nov. 2013.
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